Vincent Riccati, S.J. was born in Castel-Franco, Italy.
He worked together with Girolamo Saladini in publishing his discovery, the hyperbolic functions - although Lambert is often incorrectly given this credit. Riccati not only introduced these new functions, but also derived the integral formulas connected with them, and then, still using geometrical methods. He then went on to derive the integral formulas for the trigonometric functions. His book Institutiones is recognized as the first extensive treatise on integral calculus. The works of Euler and Lambert came later.

Saladini and Riccati also considered other geometrical problems, including the tractrix, the strophoid and the four-leaf rose introduced by Guido Grandi. His father Jacobi (after whom is named the Riccati differential equation) was one of the principal Italian mathematicians of the century, and his brother Giovanni was also a prominent mathematician.

Today we treat the hyperbolic functions as pairs of exponential functions
(ex + e-x)/2 but their inventor, the Jesuit, Vincent Riccati (son of Jacob) developed them and proved their consistency using only the geometry of the unit hyperbola x2 - y2 = 1 or 2xy = 1. Vincent Riccati followed his father's interests in differential equations which arose naturally from geometrical problems. This led him to a study of the rectification of the conics in Cartesian coordinates and to an interest in the areas under the unit hyperbola. Riccati developed the properties of the hyperbolic functions from purely geometrical considerations. He used geometrical motivation even though he was familiar with the work of Euler, who had introduced the symbol and concept of the natural number e and the function ex ten years previous to Riccati's book.

Figure 1 The trigonometric functions are taken from the unit circle

Figure 2 The hyperbolic functions are taken from the unit hyperbola

Figure 3: equivalent areas: AOK = FOH

Figure 4: AKGE = AKPN - AKHF

Riccati's development of the hyperbolic functions may be described as follows. The algebra concerning circular sectors is simple since the arc length is proportional to the area of the sector. Hyperbolas, however, do not have this property, so a different algebra is needed. The latter arises naturally from a property peculiar to hyperbolas xy = k and evident in the figure. All right triangles OAK have the same area k/2 no matter what point A is chosen on the curve. This property enables us to replace area measure by linear measure OK, thus introducing a kind of logarithm.

It can be seen from figures 3 and 4 that the following areas are equivalent:
AOK = FOH because of the property just mentioned, AOQ = QKHF by subtracting the common area OKQ, AKHF = AOF by adding AQF to each. So for any point F on the curve, the area of the sector AOF equals the area under AF. This area function depends on point F as well as on H, the projection of F on the x-axis.
Addition and subtraction of areas can then be accomplished by multiplication and division of distances along the x-axis, and vice versa.

Adding and subtracting then just involve constructing the correct proportions, i.e., finding the proper OP and OG for the proportions:
OK x OP = OG x OH which represents
adding: AKPN = AKHF + AKGEOG/OK = OP/OH which represents
subtracting: AKGE = AKPN - AKHF From these definitions of the hyperbolic functions and the logarithm operations, it is possible to derive a complete array of formulas regarding the double and triple arguments and from them to deduce other formulas for the square and cube roots. Riccati goes even further and proves the integral formulas for trigonometric as well as hyperbolic functions.

Some other innovative works of Riccati include his treatment of the location of points that divide the tangents of a tractrix and also the problem posed by Guido Grandi about the four-leaf rose.